Doubly-charged scalar in four-body decays of neutral flavored mesons

In this paper, we study the four-body decay processes of neutral flavored mesons, including $\bar K^0$, $D^0$, $\bar B^0$, and $\bar B_s^0$. These processes, which induced by a hypothetical doubly-charged scalar particle, violate the lepton number. The quantity $Br\times\left(\frac{s_\Delta h_{ij}}{m_\Delta^2}\right)^{-2}$ of different channels are calculated, where $s_\Delta$, $h_{ij}$, and $M_\Delta$ are parameters related to the doubly-charged scalar. For $\bar K^0\rightarrow h_1^+h_2^+l_1^-l_2^-$, $D^0\rightarrow h_1^-h_2^-l_1^+l_2^+$, and $\bar B_{d,s}^0\rightarrow h_1^+h_2^+l_1^-l_2^-$, it is of the order of $10^{-13}\sim 10^{-11}$ ${\rm GeV^4}$, $10^{-17}\sim 10^{-10}$ ${\rm GeV^4}$, and $10^{-17}\sim 10^{-10}$ ${\rm GeV^4}$, respectively. Based on the experimental results for the $D^0\rightarrow h_1^-h_2^-l_1^+l_2^+$ channels, we also set the upper limit for $\frac{s_\Delta h_{ij}}{M_\Delta^2}$.

In Sec. III, we give the branching ratios of all the decay channels and compare the results of D 0 with the experimental data. The last section is reserved for the conclusion. Some details for the wave functions of mesons are presented in the Appendix.

II. THEORETICAL FORMALISM
The assumed Higgs triplet ∆ in the 2 × 2 representation is defined as [10] It mixes with the usual SU(2) L Higgs doublet by a mixing angle θ ∆ , from which we define s ∆ = sin θ ∆ and c ∆ = cos θ ∆ .
The Lagrangian which describes the interaction between ∆ and W − gauge boson or SM fermions has the following form [10,13] where C = iγ 2 γ 0 is the charge conjugation matrix; ψ iL represents the leptonic doublet; h ij is the leptonic Yukawa coupling constant; g is the weak coupling constant. Compared with the second term, the third and the fourth terms can be neglected as they give smaller contributions.
If q = q 3 , all the four diagrams of Fig. 1 will contribute: where the factor 1 3 in M B and M D is introduced by the Fierz transformation; lepton is the leptonic part of the transition matrix element; V q i q j is the Cabibbo-Kobayashi-Maskawa matrix element. The definition of the decay constant f h 1 of a pseudoscalar meson is used. For vector mesons, it should be replaced by In Table I, the values of the decay constants are presented. Finally, we get the transition  [16]; K * and ρ, are from Ref. [23]; D * and D * s are from Ref. [24].
If q = q 3 , only Fig. 1(A) and (B) contribute: The hadronic transition matrix can be expressed as [25] h for h 1 being a pseudoscalar meson, where f + and f − are form factors. If h 1 is a vector meson, we have where f V and f i (i = 0, 1, 2) are form factors; M and M 1 are the masses of corresponding mesons; the definition Q = p − p 1 is used.
By applying the Bethe-Salpeter method with the instantaneous approximation [26], the hadronic matrix element is written as where ϕ ++ is the positive energy part of the wave function whose expression can be found in the Appendix; q and q 1 are the relative three-momenta between the quark and antiquark in the initial and final mesons, respectively.
The same is true for δ h 1 h 2 . The integral limits are where M 2 , m 1 , and m 2 are the masses of h 2 , l 1 , and l 2 , respectively.
Kinematics of the four-body decay of h in its rest frame. P 1 and P 2 are respectively the momenta of h 1 and h 2 in their center-of-momentum frame; P 3 and P 4 are respectively the momenta of l 1 and l 2 in their center-of-momentum frame.

III. NUMERICAL RESULTS
The wave functions of the initial and final mesons in the hadronic transition matrix element are evaluated numerically by solving corresponding instantaneous Bethe-Salpeter equation. The interaction kernel we adopt is a Cornell-type potential whose expression is presented in the Appendix. Strictly speaking, although the instantaneous approximation is reasonable for the double heavy mesons and acceptable for the heavy-light mesons, it will bring large errors for the light mesons, such as π and K. Nevertheless, we also apply this approximation to the light mesons to make the estimation, as the decay channels considered here are related to the new physics, of which only the order of magnitude is important.
This is also the reason why we do not consider the QCD corrections and the final meson interactions as mentioned in Ref.
[1]. , where Br represents the branching ratio whose upper limit can be easily achieved by using the parameter mentioned above.
ForK 0 , there are only three such channels allowed by the phase space, namely . The corresponding diagrams are Fig. 1 is of the order of 10 −11 GeV 4 (see Table II). Experimentally, Br(K + → π − l + 1 l + 2 ) 10 −10 [27], which is the most precise result of the lepton number violation decay channels. However, there is no experimental detections of lepton number violation four-body decay channels of this particle. In Refs. [28,29], the channels K L,S → π + π − e + e − are investigated. We expect there will be experiments on K L,S → π + π + l − 1 l − 2 channels. For D 0 , the final mesons can be pseudoscalars or vectors. When h 1 and h 2 are both pseudoscalars, that is ππ, πK, or KK, the results are given in Table III. The largest value has the order of magnitude 10 −10 GeV 4 . One notices that the Fermilab E791 Collaboration once presented the upper limits of the branching ratios of these channels [30], which are of the order of 10 −5 . By comparing the theoretical prediction and the experimental data, we can set the upper limit of the constant , which is of the order of 10 3 GeV −2 . One can also extract this upper limit from the three-body decay processes, such as D − → π + e − e − , which is about 10 2 GeV −2 by using the results in Ref. [10]. The decay channels of D 0 with h 1 and h 2 being 0 − 1 − or 1 − 1 − are presented in Table IV, which have the largest value about The results forB 0 andB 0 s are presented in Table V∼X. The largest value is also of the order of 10 −10 GeV 4 . In Ref. [31], the four-body decay channel B − → D 0 π + µ − µ − are measured to have the branching ratio less than 1.5 × 10 −6 . There is no experimental results for the neutral B meson decay channels until now. However, as LHCb running, more and more data will be accumulated. We expect that the LHCb Collaboration will detect such decay modes and set more stringent constraint on the parameters of doubly-charged Higgs boson. Besides that, the future B-factories, such as Belle-II, also has the possibility to provide more information about such channels.

IV. CONCLUSIONS
As a conclusion, we have studied the lepton number violation four-body decay processes of neutral flavored mesons, includingK 0 , D 0 ,B 0 , andB 0 s . They are assumed to be induced by a doubly-charged scalar. ForK 0 , the channelK 0 → π + π + e − e − has the largest value of , which is of the order of 10 −11 GeV 4 . For D 0 , the channel D 0 → π − K − l + 1 l + 2 has the largest order of magnitude 10 −10 GeV 4 . By comparing with the E791 experimental data, we set the upper limit for where q µ ⊥ = q µ − P ·q M 2 P µ , ω 1 = m 2 1 − q 2 ⊥ , and ω 2 = m 2 2 − q 2 ⊥ ; m 1 and m 2 are the masses of quark and antiquark, respectively; Λ ± i = 1 2ω i / P M ω i ± / q ⊥ + (−1) i+1 m i is the projection operator. In the above equation, we have defined and with ϕ P (q ⊥ ) being the wave function, which is constructed by / q ⊥ , / P , and polarization vector.
Here we only present the expression for the positive energy part of the wave function. For the 1 − state, it has the form For the 0 − state, it has the form A i and B i are functions of q 2 ⊥ , whose numerical results are achieved by solving Eq. (A1). The interaction potential used in this work has the form [32]